Abstract
In this paper, we establish the Gierer–Meinhardt model with cross-diffusion, and study Turing instability of its periodic solutions. Firstly, the stability of periodic solutions for the zero-dimensional system is studied by using the center manifold theory and normal form method. Secondly, according to Hopf bifurcation theorem, the diffusion rate formula for determining Turing instability of periodic solutions is established. Thirdly, by using the implicit function existence theorem and Floquet theory, the conditions of Turing instability of periodic solutions are derived, and it is proved that the periodic solutions of the model will undergo Turing instability. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is actually induced by cross-diffusion.
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